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Irrelevance in Type Theory with a Heterogeneous Equality Judgement
"... Abstract. Dependently typed programs contain an excessive amount of static terms which are necessary to please the type checker but irrelevant for computation. To obtain reasonable performance of not only the compiled program but also the type checker such static terms need to be erased as early as ..."
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Abstract. Dependently typed programs contain an excessive amount of static terms which are necessary to please the type checker but irrelevant for computation. To obtain reasonable performance of not only the compiled program but also the type checker such static terms need to be erased as early as possible, preferably immediately after type checking. To this end, Pfenning’s type theory with irrelevant quantification, that models a distinction between static and dynamic code, is extended to universes and large eliminations. Novel is a heterogeneously typed implementation of equality which allows the smooth construction of a universal Kripke model that proves normalization, consistency and decidability.
Nominal System T
, 2010
"... This paper introduces a new recursion principle for inductive data modulo ..."
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This paper introduces a new recursion principle for inductive data modulo
Extensional normalization in the logical framework with proof irrelevant equality
 In Workshop on Normalization by Evaluation, affiliated to LiCS 2009, Los Angeles
, 2009
"... We extend the Logical Framework by proof irrelevant equality types and present an algorithm that computes unique long normal forms. The algorithm is inspired by normalizationbyevaluation. Equality proofs which are not reflexivity are erased to a single object ∗. The algorithm decides judgmental eq ..."
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Cited by 2 (2 self)
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We extend the Logical Framework by proof irrelevant equality types and present an algorithm that computes unique long normal forms. The algorithm is inspired by normalizationbyevaluation. Equality proofs which are not reflexivity are erased to a single object ∗. The algorithm decides judgmental equality, its completeness is established by a PER model. 1.
Towards Normalization by Evaluation for the βηCalculus of Constructions
"... Abstract. We consider the Calculus of Constructions with typed betaeta equality and an algorithm which computes long normal forms. The normalization algorithm evaluates terms into a semantic domain, and reifies the values back to terms in normal form. To show termination, we interpret types as part ..."
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Abstract. We consider the Calculus of Constructions with typed betaeta equality and an algorithm which computes long normal forms. The normalization algorithm evaluates terms into a semantic domain, and reifies the values back to terms in normal form. To show termination, we interpret types as partial equivalence relations between values and type constructors as operators on PERs. This models also yields consistency of the betaetaCalculus of Constructions. The model construction can be carried out directly in impredicative type theory, enabling a formalization in Coq. 1
ON IRRELEVANCE AND ALGORITHMIC EQUALITY IN PREDICATIVE TYPE THEORY
"... Abstract. Dependently typed programs contain an excessive amount of static terms which are necessary to please the type checker but irrelevant for computation. To obtain reasonable performance of not only the compiled program but also the type checker such static terms need to be erased as early as ..."
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Abstract. Dependently typed programs contain an excessive amount of static terms which are necessary to please the type checker but irrelevant for computation. To obtain reasonable performance of not only the compiled program but also the type checker such static terms need to be erased as early as possible, preferably immediately after type checking. To this end, Pfenning’s type theory with irrelevant quantification, that models a distinction between static and dynamic code, is extended to universes and large eliminations. Normalization, consistency, and decidability are obtained via a universal Kripke model based on algorithmic equality. 1. Introduction and Related
On ηExpansion in NbE and Type Casts
, 2011
"... This small note justifies the asymmetry in the definition of the ηexpansion functions ↑ and ↓ for function types in the context of normalization by evaluation for dependent types [ACD07, Abe10, ACP11]. ( ↑ Fun A F n) a = ↑ F a n ( ↓ A a) ( ↓ Fun A F f) x = ↓ F ↑A x f (↑ A x) The asymmetry F a vs. ..."
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This small note justifies the asymmetry in the definition of the ηexpansion functions ↑ and ↓ for function types in the context of normalization by evaluation for dependent types [ACD07, Abe10, ACP11]. ( ↑ Fun A F n) a = ↑ F a n ( ↓ A a) ( ↓ Fun A F f) x = ↓ F ↑A x f (↑ A x) The asymmetry F a vs. F ↑ A x can be derived from a view of ↑ and ↓ as embeddingprojection pair, or up and downcast functions, with a symmetric definition: ( ↑ Fun A ′ F ′ Fun A F f) a ′ = ↑ F ′ a ′
A TypeChecking Algorithm for MartinLöf Type Theory with Subtyping Based on Normalisation by Evaluation
"... Abstract. We present a core MartinLöf type theory with subtyping; it has a cumulative hierarchy of universes and the contravariant rule for subtyping between dependent product types. We extend to this calculus the normalisation by evaluation technique defined for a variant of MLTT without subtyping ..."
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Abstract. We present a core MartinLöf type theory with subtyping; it has a cumulative hierarchy of universes and the contravariant rule for subtyping between dependent product types. We extend to this calculus the normalisation by evaluation technique defined for a variant of MLTT without subtyping. This normalisation function makes the subtyping relation and typechecking decidable. To our knowledge, this is the first time that the normalisation by evaluation technique has been considered in the context of subtypes, which introduce some subtleties in the proof of correctness of NbE; an important result to prove correctness and completeness of typechecking. 1